LAS VEGAS — Sen. Ted Cruz (R-Tex.) wanted to barnstorm Nevada as the winner of the Iowa caucuses, the only candidate who “has defeated Donald Trump” and the one who could do it again.

Instead, Cruz spent some of the scant hours Monday between Saturday’s South Carolina primary and this state’s caucuses firing a well-liked spokesman, Rick Tyler, who had shared a Facebook video of Sen. Marco Rubio (R-Fla.) appearing to disparage the Bible. It was a dramatic attempt to end accusation...